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| Авторизация |
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| Wall H.S. — Analytic Theory of Continued Fractions |
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| Предметный указатель |
Abel 36
Approximant 15
Approximant as Stieltjes transform 254
Approximant of positive definite continued fraction 72
Approximant, elements in parabolic domains 136
Approximant, values in circular domain 42 46 60
Arzela 104
Associative law for bounded matrices 221
Asymptotic equality 316
Asymptotic equality between J-fraction and equivalent function 318
Asymptotic equality between power series and J-fraction 321
Asymptotic equality in angular domain 324
Asymptotic equality on imaginary axis 324
Bessel function 349
Beth 235
Bilinear form 216
Bilinear form, completely continuous 236
Blaschke 105
Bounded analytic functions in unit circle 283 285
Bounded analytic functions, chain sequences 279ff.
Bounded analytic functions, imaginary part negative 299
Bounded analytic functions, integral formulas for 275
Bounded analytic functions, real part positive 279 288
Bounded analytic functions, reciprocal of 280
Bounded J-fraction 119ff.
Bounded J-fraction, bound 111
Bounded J-fraction, convergence of 112 114
Bounded J-fraction, convex set for 113
Bounded J-fraction, norm 111
Bounded transformation 220
Bray 239
C-fraction 400
C-fraction,  -regular 409
C-fraction, examples 403—405
C-fraction, regular 405
Cardioid theorem 142
Carleman 109 238 330
Carleman’s inequality 330
Carleman’s theorem on the moment problem 330
Cesaro mean 310
Chain sequence 79
Chain sequence, bounded analytic functions 278 288
Chain sequence, constant term 79
Chain sequence, maximal parameters positive 82
Chain sequence, minimal and maximal parameters 80
Chain sequence, moment problem for (0,1) 263
Chain sequence, parameters determined uniquely 82
Chain sequence, parameters of 79
Chain sequence, quadratic forms 86
Chain sequence, real J-fraction of norm  115
Computation of roots of polynomials 185
Condition (H), necessary for convergence 122
Condition (H), theorem on sufficiency of 123
Continued fraction equivalent to a series 17
Continued fraction for certain divergent power series 350
Continued fraction of Gauss 337
Continued fraction of Gauss, convergence of 339
Continued fraction of Gauss, inequalities for 342
Continued fraction, expansions for functions bounded in unit circle 285
Continued fraction, expansions for functions in Ext  279 288
Continued fraction, expansions for rational functions 161
Continued fraction, generated by linear transformations 14
Continued fraction, of Gauss 337
Continued fraction, of Pade table 380
Continued fraction, positive definite 67
Continued fraction, special type 45 48 64 103 116 118 285 289
Contraction formulas 22
Convergence continuation theorem 108
Convergence neighborhood 62 137
Convergence of continued fraction in cardioid domain 142
Convergence of continued fraction in the determinate case 110
Convergence of continued fraction in the indeterminate case 103 104
Convergence of continued fraction of Stieltjes 120
Convergence of continued fraction, condition (H) 123
Convergence of continued fraction, definition of 16
Convergence of continued fraction, exterior to cut 138
Convergence of continued fraction, necessary condition for 28 29 33
Convergence of continued fraction, periodic 37
Convergence of continued fraction, sufficient condition for 34 35 42 45 48 49 51 57 58 62 72 103 104 112 114 115 116 120 123 131 137 138 142 143 209 339 350 351
Convergence of continued fraction, uniform 16 42 45 48 49 131
Convergence of continued fraction, when partial numerator is zero 26 42 137
Convergence of continued fraction, whose elements have limits 138 139 339
Convergence of even (odd) part 53
Convergence of positive definite J-fraction, determinate case 110
Convergence of positive definite J-fraction, indeterminate case 104
Convergence of Stieltjes type continued fraction 120 123 131
Convergence problem 16
Convergence problem, general remarks on 25ff.
Convergence problem, nature of 27
Convergence set 45
Convergence set in parabolic domains 142
Convergence set, symmetric to real axis 58
Convergent see “Approximant”
Convex extension 265
Convex set 113 137 237
Convex set, containing roots of polynomial 176
Convexity 264
de la Vallee Poussin mean 280
Dennis 51
Denominator 15
Denominator, different from zero 41 137
Denominator, expressed as determinant 65
Denominator, inequalities for 55
Determinant formula 16
Determinate case 99
Determinate case, convergence in the 110
Determinate case, sufficient conditions for 109
Determinate case, uniqueness of bounded reciprocal 234
Differential equation, Laguerre’s 412
Differential equation, power series solution 360 361
Dines 263
Divergence of continued fraction 16
Divergence of continued fraction, sufficient conditions for 27 28 29 33
Elements 14
Equivalence transformation 19
Equivalent function of J-fraction 231
Equivalent function of J-fraction, estimates for 235
Equivalent function of J-fraction, proper 254
Equivalent function of J-fraction, represented by Stieltjes transform 253
Euler 18
Euler — Knopp mean 309
Evans 239
Even part of continued fraction 20
Even part of continued fraction, convergence of 53
Expansion algorithm for test-fraction 175
Expansion algorithm of Stieltjes 203
Expansion algorithm, power series into C-fraction 401
Expansion algorithm, power series into J-fraction 198
Expansion algorithm, rational function into J-fraction 163
Expansion algorithm, rational function into S-fraction 171
Extension formulas 22
Formal integration 192
Forms, bilinear 216
Forms, linear 214
Forms, quadratic 222
Forms, unit 216
Frank 178 400
Frobenius 378
Functions of bounded variation 241
Functions with natural boundary 414
Functions, analytic exterior to cut 209
Functions, Bessel 349
Functions, binomial 343
Functions, elliptic 211 374 375
Functions, equivalent of J-fraction 231
Functions, error 357 358
Functions, exponential 348 349 367
Functions, hyperbolic 206 366 369—372
Functions, hypergeometric 335
Functions, incomplete gamma 356
| Functions, logarithmic 200 342 343 344
Functions, meromorphic 209 347
Functions, nondecreasing 246
Functions, proper equivalent functions 254
Functions, psi function 372 373
Functions, quasi-analytic 330
Functions, represented by Stieltjes transform 247 253 254
Fundamental inequalities 40
Fundamental inequalities, first interpretation 42
Fundamental inequalities, second interpretation 54
Fundamental recurrence formulas 15
Gamma function, incomplete 356
Gamma function, Stieltjes continued fraction for 365
Green’s formula 96
Grommer 238
Gronwall 280
hadamard 399 412
Hadamard gaps 411
Hamburger 29 94 101 123 132 238 321
Hausdorff 761 111
Hausdorff inclusion problem 311
Hausdorff means 304
Hausdorff means, convergence preserving 307
Hausdorff means, examples of 309
Hausdorff means, form of 306
Hausdorff means, regular 309
Hellinger 69 95 96 98 109 214 233 235 238
Herglotz 275
hilbert 214 238 245
Hilbert space 215
Hilbert sphere 215
Hitchcock 185
Holder mean 310
Hypergeometric series 335
Indeterminate case 99
Indeterminate case, convergence theorem for 101
Inequalities, for denominators 55
Inequalities, for roots of polynomials 177
Inequalities, fundamental 40
Integral representation of analytic functions 275
J-form 103
J-form with five diagonals 334
J-form, bounded 111
J-form, norm 111
J-fraction 103
J-fraction for rational function 161
J-fraction for rational function, condition for existence 165
J-fraction for rational function, expansion algorithm 163
J-fraction for rational function, relation to power series 166
J-fraction, bounded 110ff
J-fraction, equal to power series 208
J-fraction, equivalent function of 231
J-fraction, even part of Stieltjes continued fraction 119
J-fraction, formally equal to  213
J-fraction, indeterminate case 103
J-fraction, moments of 197
J-fraction, positive definite 104
J-fraction, power series expansion of 197
J-fraction, real 114
J-matrix 226
J-matrix for positive definite J-fraction 230
J-matrix, arbitrary 229
J-matrix, bounded 226
J-matrix, uniqueness of bounded reciprocal 234
Laguerre 412
Laguerre’s differential equation 412
Lambert’s continued fraction 349
Lane 36 61 91 148
Least squares 211
Legendre polynomials 344
Leighton 27 31 400
Limit-point and limit-circle cases 73
Linear forms 214
Linear systems 223
Linear transformations in infinitely many variables 212
Linear transformations in unit circle 286
Linear transformations, bounded 220
Linear transformations, fixed points of 36
Linear transformations, product of 13 36
Majorant series 41
Majorant series, divergent 54
Mall 29 31 132
Matrices, bounded 218
Matrices, Cesaro 301
Matrices, completely continuous 236
Matrices, Hausdorff 304
Matrices, J-matrices 226
Matrices, norm of 218
Matrices, permutable 302
Matrices, product 219
Matrices, reciprocal (right, left) 223
Matrices, sum 219
Matrices, symmetric 222
Matrix calculus 214
Moment problem 258
Moment problem for a finite interval 258 271
Moment problem for a finite interval, geometric characterization 265
Moment problem for a finite interval, normalized solution 259
Moment problem for a finite interval, solution by S-fractions 260
Moment problem for a finite interval, symmetrical 259
Moment problem for a finite interval, totally monotone sequences 267
Moment problem for an infinite interval 325
Moment problem for an infinite interval, Carleman’s theorem on 330
Moment problem for an infinite interval, determinate and indeterminate 326
Moments of J-fraction 197
Montel 105
Montessus de Balloire 412
Nest of circles  71
Nest of circles , radius  74—75
Nevanlinna, F. 334
Nevanlinna, R. 238 321 324
Norm, J-form 111
Norm, J-fraction 111
Norm, matrix 218
Notation for continued fraction 17
Numerator 15
Odd part of continued fraction 21
Odd part of continued fraction, convergence of 53
Orthogonal polynomials 193
Osgood 104
pade 378
Pade approximant 378
Pade approximant, degree of 388
Pade table 378
Pade table, block of order r 394
Pade table, diagonal files 393
Pade table, for series of Stieltjes 389
Pade table, general theorems on 393
Pade table, horizontal files of 412
Pade table, normal 379 388 399
Pade table, square in 389
Pade table, stairlike files in 380
Parabola theorem 58
Parabola theorem, connection with positive definite continued fraction 75
Parabola theorem, extension of 142
Parabola theorem, question of uniform convergence 78
Parabolic domains 58
Parabolic domains, family of 135 152
Partial denominator 14
Partial denominator, equal to zero 27
Partial fraction development, real J-fractions 115
Partial fraction development, Stieltjes continued fractions 120
Partial numerator 14
Partial numerator, equal to zero 26 42 72
Partial quotient 14
Paydon 44 48 79 137
Periodic continued fraction 23
Periodic continued fraction, convergence and value 37 39 190
Perron 22 36 48 50 137 412
Polynomials with real roots 172
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